OF AIR OBSERVED IN KUNDT’S TUBES. 
19 
or as we may take it approximately, if y x be small compared with the wave-length X, 
i|)= iam^ {A , (2/i _ 3/) + B , (% _ 2/)3j . (86) . 
The value of cr to a second approximation would have to be investigated by means 
of (62). It will be composed of two parts, the first independent of t, the second a 
harmonic function of 2 nt. In calculating the part of dfyjdx independent of t from 
0 da da da 
V a <p= — ~rr — 
T dt dx dy 
we shall obtain nothing from d<r/dt. In the remaining terms on the right-hand side it 
will be sufficient to employ the values of u, v, a- of the first approximation. From 
in conjunction with (80), we get 
da du dv 
dt dx dy 
■u n . , 
<t ——— sm kx sin nt, 
whence 
d-<b kud 9 . 
mr=2^- cos kxe ^ sm ^- 
It is easily seen from this that the part of u resulting from d<f>/dx is of order P/3 3 in 
comparison with the part (87) resulting from xp, and may be omitted. 
Accordingly by (84), with introduction of the value of /3 and (in order to restore 
homogeneity) of u {j 2 
-ltd 2 sin 2kxe~^, M . _ . „ _ „ . 
u— -—-{4 sm f3y-\- 2 cos (3y-\-e 
. . . . (87), 
and from (86) 
V — 
U — 
2te 0 2 cos 2 lexer 
8 ft a 
(sin /3y-f 3 cos fiy-\-^e 
sin 2 kx 
8/3a 
{A'+3B \ yi -yf} 
( 88 ); 
(89), 
2 kud cos 2 kx 
s/3a 
(90). 
When y— 0, the complete values of u and v, as given by the four last equations, 
must vanish. Determining in this way the arbitrary constants A' and B', we get as 
the complete values at any point, 
D 2 
